# Prime Solver

## A fundamental problem had stymied mathematicians for centuries. Then a former Subway cashier announced he’d solved it.

###### Photos by: Winslow Townson

What does a eureka moment look like? It’s usually more subtle than having an apple fall from a tree, striking a young scientist in the head. In fact, you might watch the whole thing unfold without realizing anything took place at all.

That’s what happened on July 3, 2012, when a middle-aged Chinese man passed a seemingly uneventful half-hour in the backyard of his friend’s suburban home in Pueblo, Colo. The man, Yitang Zhang, spent most of the time pacing, occasionally glancing at the golf course that abutted the yard or gazing toward Pikes Peak and other mountains in the distance. He didn’t say a word, and after 30 minutes he went inside the house, perhaps to find relief from the heat and blazing sun.

Zhang had just solved one of the most celebrated problems in mathematics, a version of the twin prime conjecture. (A conjecture’s a mathematical hypothesis with some basis to it, not just a wild guess.) He didn’t bother telling his friend what he’d accomplished, as it would have been rather difficult to explain, even if Zhang was inclined to talk about his work, which he’s not. The effort of trying to get a layperson to grasp the miraculous, though exceedingly technical, sleight of hand he’d just pulled off would likely have ruined the moment of private exaltation Zhang was experiencing. “I didn’t tell him anything,” he says. “It was unnecessary to tell him.”

Yitang Zhang on the University of New Hampshire campus in Durham.

After making his great intellectual leap, Zhang didn’t bother writing anything down either, as he hadn’t brought a computer or even a pad of paper on his trip. Instead, he kept the ideas locked up in his head and continued his vacation for another four days, as if nothing out of the ordinary had transpired. “Everything was in my brain,” he says. “That is my way as I usually do not talk much to other people.” He can talk fluently when he has to, and is always polite, but he exudes a bit of impatience as if to suggest he has something else on his mind.

Zhang had just solved one of the most celebrated problems in mathematics, a version of the twin prime conjecture.

On this particular occasion, Zhang had plenty on his mind, although he betrayed little to those who saw him. It was not until he returned to his office at the University of New Hampshire, where he was a lecturer in mathematics, that he got the chance to follow up on his dramatic insight and start writing the proof. It took him about nine months to flesh out all the details and submit his paper, on April 17, 2013, to the Annals of Mathematics.

Within three weeks, an unusually fast time, he received some rave reviews from the paper’s referees, who determined that his argument was airtight. The journal agreed to publish Zhang’s work, and word of his feat quickly spread through the math world and well beyond. As a result, the hitherto unknown mathematician became famous almost overnight.

### Prime Time

So what exactly did he prove? His argument relates to the distribution of prime numbers, which are whole numbers (not counting one) with no factors other than one and themselves. Prime numbers are the foundation of our numbering system, as every positive integer greater than 1 is either a prime number or the product of primes. The twin prime conjecture looks at the gaps, or spacing, between prime numbers, focusing specifically on pairs of primes — so-called “twin primes” — that differ by just two (3 and 5, for example, or 1,000,000,007 and 1,000,000,009).

As numbers get bigger, primes become sparser. But Euclid proved some 2,300 years ago that there are an infinite number of primes. So, what about the gaps between primes? Can they ever become infinitely large?

The twin prime conjecture, which dates back to the 1800s and had seen scant progress in the 150 years since, says that no matter how far up the number line you go, you will never run out of twin primes, even though the average distance between prime numbers grows. The largest twin primes identified to date are 3,756,801,695,685 x 2666,669 - 1 and 3,756,801,695,685 x 2666,669 + 1, numbers with 200,700 digits each. If the conjecture is right, then no matter how big numbers get, there will always be bigger twin pairs to be found.

He was the first to show that gaps between successive primes do not grow indefinitely, even as prime numbers themselves become fewer and farther between. It’s an amazing result, made all the more unusual by its unlikely origins.

But that, it turns out, has been hard to prove. In fact, the twin prime conjecture had “earned the reputation among most mathematicians in the field as hopeless in the sense that there is no known unconditional approach for tackling it,” according to a 2005 paper written by mathematicians Daniel Goldston, János Pintz and Cem Yildirim. Goldston was stunned upon first learning of Zhang’s paper, saying he initially assumed that it must be wrong. “The whole thing seemed unbelievable,” he says.

Yitang Zhang, now a full professor at UNH, teaches students abstract algebra.

While Zhang didn’t solve the twin prime conjecture itself, he did prove what many mathematicians consider to be the essential part of it — namely that the distance between successive primes always remains finite. More specifically, he developed an advanced prime-finding method to show that there are infinitely many pairs of primes that differ by less than a fixed constant, H. In his proof, H had the value of 70 million, but that’s just an upper bound; he left the door open for smaller values. (The actual twin prime conjecture sets H at 2.) In this way, he was the first to show that gaps between successive primes do not grow indefinitely, even as prime numbers themselves become fewer and farther between. It’s an amazing result, made all the more unusual by its unlikely origins.

### From Zero to Hero (Sandwich)

Zhang’s paper, “Bounded gaps between primes,” was a shocker on many levels. It’s rare, almost unheard of in mathematics, for a major result to come out of the blue like this. Given his reticence during the many months of unaccompanied labor, Zhang says, “No one could see it coming.” Practically nobody had heard of him before or had any clue that he was even working on this problem, as he hadn’t published anything on the subject. In fact, he’d only published twice before in his entire career — his master’s thesis in 1985 and another paper in 2001 — both relating to another well-known conjecture called the Riemann hypothesis. Neither of those papers got much attention.

Born in China in 1955, Zhang was a top student at Peking University who harbored a keen interest in number theory — the study of non-negative integers and the relations between them. He got his master’s degree in 1985 and hoped to continue his graduate studies at the University of California, San Diego, where another Chinese mathematician had arranged for him to work under the guidance of a well-respected number theorist. But the president of Peking University prevented it, urging Zhang to study algebraic geometry at Purdue because he felt that Chinese mathematicians were weak in that subject.

Zhang had absolutely no interest in the thesis problem his Purdue adviser forced upon him. “I worked on it against my will,” he insists. Although he managed to get his Ph.D. in 1991, he did not receive a letter of recommendation from his adviser, making it difficult for him to find an academic position in mathematics. For the next five or so years, he lived in Kentucky, helping a friend who ran a chain of Subway fast-food restaurants. By his own admission, Zhang wasn’t great at making sandwiches, though he could lend a hand in a crunch. He was, however, a standout cashier, as well as a credible accountant. All his work with numbers had not gone to waste.

During slow periods on the job, Zhang’s thoughts invariably turned toward mathematics. “If you love math, you will find time to do it, despite the difficult environment,” he says. It wasn’t always easy, but he tried to follow his passion whenever he had the chance.

In 1999, he caught a break. A friend from Peking University on the UNH faculty helped Zhang get a non-tenure-track job in the math department teaching courses and filling in as a substitute. When he was not in the classroom, he spent a fair amount of time in the hallways, seemingly lost in thought. For months during this period, a visiting professor at UNH only knew Zhang as some sort of curious phantom figure, appearing in the corridors without notice, pacing around for a while and then disappearing just as suddenly. It was not until the spring of 2013 — when Zhang published his “bounded gaps” paper — that UNH mathematicians (and the rest of the world) found out what their silent colleague had been preoccupied with for so long.

Zhang never bought into the notion that solving the twin prime conjecture was hopeless. “I just keep going,” he says. “My approach to attacking a problem like this is to keep thinking” — a tactic he applies no matter where he is, and regardless of his state of wakefulness. “Sometimes, I even get new ideas in a dream,” he says. Zhang leads a simple life by design, one geared toward quiet contemplation. He shares a modest apartment with a graduate student in Dover, N.H., about five miles from the UNH campus in Durham. He has no car, no TV and few distractions from his work. “I’ve found that when I watch a TV show, I could not concentrate on the TV itself,” he says. “My mind wanders to mathematics.” And that single-mindedness might have helped him succeed where so many before him had gotten stuck.

### Heart on His Sieve

Like many problems in number theory, the twin prime conjecture appealed to him because the idea could be stated so simply, even if the solution was anything but. He started working on the problem in earnest around 2009, when he came across the paper by Goldston, Pintz and Yildirim. GPY, as the trio was called, sketched out a path for solving the problem and made considerable progress, but they couldn’t reach the finish line. Zhang got inspired while reading the paper, thinking, “Maybe I could do something with this.” But for three years, he was unable to get anywhere, running into one roadblock after another.

He grappled with this problem all the time, including during his 2012 vacation to Colorado. Somehow, Zhang found a way to proceed through the final logjam while killing time in his friend’s backyard in Pueblo. During his inconspicuous eureka moment, he recalls, he kept asking himself, “ ‘Why not go this way? Why not? Why not?’ And I concluded, ‘Yes we can!’ ” It all came down to working out one tiny step in a much longer logical argument. With that part in hand, he says, “I knew how all the other pieces fell into place.”

The key to his breakthrough was figuring out how to modify a mathematical tool called a sieve, which could be used for spotting pairs of prime numbers. The idea of a sieve for sorting numbers dates back at least 2,200 years to the Greek mathematician Eratosthenes, who developed a simple but effective method for picking out primes: Start, for example, with a list of numbers from 1 to 100. Cross out 1, because it is not a prime by definition. Take 2, which is a prime, and cross out all multiples of it. Then take 3, the next prime number, and cross out all multiples of it, too. Since 4 is a multiple of 2, it’s already been eliminated, so now cross out all multiples of 5 and 7 and so forth.

Extending that approach can identify primes well beyond 100, of course, but the method is far too slow and unwieldy to solve theoretical problems, like the twin prime conjecture, which can run into truly enormous numbers. Zhang opted for a “less selective” strategy that does not try to weed out all the multiples but instead eliminates a narrower set of multiples. This revised sieve, he says, “does not get every prime, but still gets enough of them to prove the case.” And what he proved, again, is that one can always find consecutive prime numbers that differ by less than 70 million.

### Infinity > 70 Million > 2

“Seventy million may seem like a big number to some people,” Zhang says, “but it is very small compared to infinity. I just needed one number that satisfied all the conditions.” That one fit the bill, though he knew upfront that it could be whittled down substantially.

In fact, that’s already happened. In June 2013, UCLA mathematician Terence Tao initiated the “polymath8 project” in the hopes of reducing the fixed constant H from 70 million to as low a value as possible — ideally all the way down to 2, which would prove the twin prime conjecture outright. Other mathematicians joined Tao in this endeavor and, as of July 2014, H has been cut to 246, with further reductions still possible. But neither Zhang nor Tao thinks this general approach can take us all the way to 2. That will require a new idea altogether, Zhang says. But he’s content, for now, with having established the general case — the existence of a finite bound to the gaps between primes — which he considers to be the crux of the problem anyway.

That attitude makes sense to Goldston, the “G” in GPY, who says, “There is nothing special about the number 2. The most important thing is showing that primes never get isolated — showing that there will always be primes that are close together.”

Yitang Zhang with UNH grad student Mahtab Lak engage in one of Zhang's favorite pastimes: thinking about complex mathematical problems.

### A Mind for Math

Zhang, for his part, has already moved on to “another important, unsolved problem in number theory” related to the Goldbach conjecture, which holds that every even number is the sum of two primes.

That’s about all he’ll say for now, because he doesn’t like talking about the problems he’s working on, or even conferring with other experts. All he really wants to do is devote himself to solving these problems with minimal intrusion from the outside world. He’s got somewhat less time to do so now that he has gained some celebrity and is asked to give lectures or speak with journalists. But despite the hectic schedule and frequent travel, “mathematics is never far from my thoughts,” he says, “and hopefully fame will not change that.”

At least his ability to pursue mathematical research is more secure now than it was in the 1990s, when he was forced to make sandwiches and run cash registers to pay the bills. UNH recently granted Zhang a full professorship, and he’s received job offers from universities in China, Taiwan and elsewhere. In 2014 he won a MacArthur “Genius” Grant, and spent the first half of the year as a visiting professor at the Institute for Advanced Study in Princeton, N.J. (Einstein’s former home). In addition, he’ll do research at the Chinese Academy of Sciences in Beijing two months every year, but beyond that his long-term plans remain unsettled.

Whether he stays in Durham or goes elsewhere may not make a huge difference to Zhang, as long as he ends up in a situation that allows him to do what he loves most — to think about vexing problems in number theory to his heart’s content, at any time of day or night, no matter what’s happening around him. And, when he’s done with that, to think some more.